# proof that all cyclic groups are abelian

###### Proof.

Let $G$ be a cyclic group and $g$ be a generator of $G$. Let $a,b\in G$. Then there exist $x,y\in{\mathbb{Z}}$ such that $a=g^{x}$ and $b=g^{y}$. Since $ab=g^{x}g^{y}=g^{x+y}=g^{y+x}=g^{y}g^{x}=ba$, it follows that $G$ is abelian. ∎

Title proof that all cyclic groups are abelian ProofThatAllCyclicGroupsAreAbelian 2013-03-22 13:30:44 2013-03-22 13:30:44 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Proof msc 20A05